At the beginning of the war a lot of the regiments probably wore their civilian clothing. There was no standing army in 1642 and with the exception of the trained bands (militia) regiments, both armies had to be built up more or less from scratch. Uniform coats were issued in large numbers on a regimental basis, it generally being the responsibility of the colonel to cloth his own regiment. What those coats actually looked like is a matter of huge debate, there are no surviving examples and only imperfect depictions. Some regiments issues other clothing such as breeches and caps. In 1643 the King granted contracts to clothe his army in Oxford, and the prevailing opinion is that the clothing was red or blue. For Parliament the New Model Army was the first real attempt at centralised uniform.

The important thing is that the clothing of both sides was the same. Visually, there was really no outward signifier that distinguished one side from the other.

There is an ongoing project researching military units of the English Civil War, with some information on uniform colours, here: BCW Project

The only uniform item were "favours" worn by the two armies, which very quickly emerged after the start of the war. Parliament used what was described as "orange tawny" and the King used "crimson red" but these weren't universal, and other favours could be used, like bits of paper in hatbands, or other tokens to indicate army and allegiance. This "The English Civil War, 1642-1651: An Illustrated Military History" by Philip J. Haythornthwaite is a good guide.